Almost critical local well-posedness for the space-time Monopole equation in Lorenz gauge

Abstract

Recently, Candy and Bournaveas proved local well-posedness of the space-time monopole equation in Lorenz gauge for initial data in $H^s$ with $s>\frac{1}{4}$. The equation is $L^2$-critical, and hence a $\frac{1}{4}$ derivative gap is left between their result and the scaling prediction. In this paper, we consider initial data in the Fourier-Lebesgue space $\widehat{H_p^s}$ for $1<p\le 2$ which coincides with $H^s$ when $p=2$ but scales like lower regularity Sobolev spaces for $1<p<2$. In particular, we will see that as $p\to 1^{+}$, the critical exponent $s_p^c\to 1^{-}$, in which case $\widehat{{\dot{H}}_{1+}^{1-}}$ is the critical space. We shall prove almost optimal local well-posedness to the space-time monopole equation in Lorenz gauge with initial data in the aforementioned spaces that correspond to $p$ close to 1.

arXiv comment15 pages